That kind of thing would be a judgment call on the part of the crew (ultimately the capt) and would come down to what is the safest action based on the circumstances, with logistical/convenience considerations being a distant second.
When you have a blown tire you will want to avoid raising the gear, because they can catch fire and you don't want to retract the fire into the fuselage (and your own fire detection system won't tell you this until it's retracted, so you really won't know until it's too late). On top of that, you may have engine damage, especially if they are tail mounted, and/or flap damage.
So the imperative is to get the hell back down post-haste. However, the airport is also a consideration. Most of the time the departure airport has long runways and good emergency services, so the crew will elect to do a turnback even if fuel dumping is necessary. In other circumstances, say if the departure airport is remote and has short runways with minimal emergency facilities, the crew may elect to divert to the nearest airport with suitable facilities, and will likely do the diversion leaving the gear down.
Individual airlines may have specific policies on this as well, and a capt at one airline may have more discretion on what to do than another.